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21    PRODUCT OF MEASURES        221

for each pair of indices /, k, and such that /X*) ^ 1 for all x e X,

for all j; e Y, and /,(*) = 1 for all x e S,


Then we have

= 1 for all y e T (12.6.4).

for all x e X and all y e Y. If yk is any point of Bfc , the hypotheses there-
fore imply that

, y) -

for all (x, j) e A x B, where A = \J At and B = (J Bfc , and hence also for all
(x, y) e X x Y (since the expression on the left vanishes outside A x B).

(2)    Existence.    First we shall prove the following lemma:

(   Let L be a compact subset of X and M a compact subset
0/Y. If he tfc(X x Y) is such  that Supp(/z) c L x M,  then the function

g(y) =    h(x, y) dX(x) is continuous on Y, and Supp(^) cz M.

For each jeY, the function x\-+h(x,y) belongs to JTC(X; L) and is
identically zero if jy ^ M. On the other hand, because h is uniformly continuous
(3.16.5), for each e > 0 and each y e Y there exists a neighborhood W of y in
Y such that the relations x e X and / e W imply that \h(x, y')  h(x, y)\ <j e.
Finally, there exists a number aL > 0 such that |A(w)| ^ tfiJMI fr a^
u e Jf C(X; L). We deduce that


1000 - 0001 =

for all y' e W, and the lemma is proved.

We return to the proof of (13.21.1). For each function h e ^TC(X x Y), the
number v(h) = n(g) (which by abuse of notation is also written in the form

p( fh(x, y) dk(x))j is defined. Furthermore, with the notation used above,
there exist by hypothesis two numbers aL and bM such that,  for  eachh in U x V is :ge